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Well posedness and stationary solutions of a neural field equation with synaptic plasticity

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This paper is available in a repository.

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Abstract

We consider the initial value problem associated to the neural field equation of Amari type with plasticity \[ u_t(x,t)=-u(x,t)+∫_{Ω}w(x,y)[1+γ g( u(x,t) - u(y,t) )] f(u(y,t))\; dy, \;(x,t) 𝟄 Ω \times (0, ∞), \] where $Ω⊂ℝ^m$, $f$ and $g$ are bounded and continuously differentiable functions with bounded derivative, and $γ\ge0$ is the plasticity synaptic coefficient. We show that the problem is well posed in $C_b(ℝ^m)$ and $L^1(Ω)$ with $Ω$ compact. The proof follows from a classical fixed point argument when we consider the equation's flow. Strong convergence of solutions in the no plasticity limit ($γ\to0$) to solutions of Amari's equation is analysed. Finally, we prove existence of stationary solutions in a general way. As a particular case, we show that the Amari's model, after learning, leads to the stationary Schr̈odinger equation for a type of gain modulation.